System and methods for simultaneous momentum dumping and orbit control

ABSTRACT

The present system and methods enable simultaneous momentum dumping and orbit control of a spacecraft, such as a geostationary satellite. Control equations according to the present system and methods generate accurate station-keeping commands quickly and efficiently, reducing the number of maneuvers needed to maintain station and allowing station-keeping maneuvers to be performed with a single burn. Additional benefits include increased efficiency in propellant usage, and extension of the satellite&#39;s lifespan. The present system and methods also enable tighter orbit control, reduction in transients and number of station-keeping thrusters aboard the satellite. The present methods also eliminate the need for the thrusters to point through the center of mass of the satellite, which in turn reduces the need for dedicated station-keeping thrusters. The present methods also facilitate completely autonomous orbit control and control using Attitude Control Systems (ACS).

BACKGROUND

1. Technical Field

The present disclosure relates to station-keeping for synchronoussatellites.

2. Description of Related Art

With reference to FIG. 1, a synchronous satellite 10 orbits the Earth 12at a rate that matches the Earth's rate of revolution, so that thesatellite 10 remains above a fixed point on the Earth 12. FIG. 1illustrates the satellite 10 at two different points A, B along itsorbit path 14. Synchronous satellites are also referred to asgeostationary satellites, because they operate within a stationaryorbit. Synchronous satellites are used for many applications includingweather and communications.

Various forces act on synchronous satellites to perturb their stationaryorbits. Examples include the gravitational effects of the sun and themoon, the elliptical shape of the Earth and solar radiation pressure. Tocounter these forces, synchronous satellites are equipped withpropulsion systems that are fired at intervals to maintain station in adesired orbit. For example, the satellite 10 illustrated in FIG. 1includes a plurality of thrusters 16.

The process of maintaining station, also known as “station-keeping”,requires control of the drift, inclination and eccentricity of thesatellite. With reference to FIG. 1, drift is the east-west position ofthe satellite 10 relative to a sub-satellite point on the Earth 12.Inclination is the north-south position of the satellite 10 relative tothe Earth's equator. Eccentricity is the measure of the non-circularityof the satellite orbit 14, or the measure of the variation in thedistance between the satellite 10 and the Earth 12 as the satellite 10orbits the Earth 12. Satellite positioning and orientation is typicallycontrolled from Earth. A control center monitors the satellite'strajectory and issues periodic commands to the satellite to correctorbit perturbations. Typically, orbit control is performed once everytwo weeks, and momentum dumping is performed every day or every otherday.

Current satellites are either spin-stabilized or three-axis stabilizedsatellites. Spin-stabilized satellites use the gyroscopic effect of thesatellite spinning to help maintain the satellite orbit. For certainapplications, however, the size of the satellite militates in favor of athree-axis stabilization scheme. Some current three-axis stabilizedsatellites use separate sets of thrusters to control north-south andeast-west motions. The thrusters may burn a chemical propellant orproduce an ion discharge, for examples to produce thrust. Alternatively,the thrusters may comprise any apparatus configured to produce avelocity change in the satellite. The north-south thrusters produce therequired north-south change in satellite velocity, or ΔV, to controlorbit inclination. The east-west thrusters produce the required combinedeast-west ΔV to control drift and eccentricity. As the cost of satellitepropulsion systems is directly related to the number of thrustersrequired for station keeping, it is advantageous to reduce the number ofthrusters required for satellite propulsion and station keeping.Further, propulsion systems have limited lifespans because of thelimited supply of fuel onboard the satellite. Thus, it is alsoadvantageous to reduce fuel consumption by onboard thrusters so as toextend the usable life of the satellite.

SUMMARY

The embodiments of the present system and methods for simultaneousmomentum dumping and orbit control have several features, no single oneof which is solely responsible for their desirable attributes. Withoutlimiting the scope of this system and these methods as expressed by theclaims that follow, their more prominent features will now be discussedbriefly. After considering this discussion, and particularly afterreading the section entitled “Detailed Description”, one will understandhow the features of the present embodiments provide advantages, whichinclude a reduction in the number of maneuvers needed to maintainstation, increased efficiency in propellant usage, reduction intransients, tighter orbit control, which has the added benefit ofreducing the antenna pointing budget, a reduction in the number ofstation-keeping thrusters needed aboard the satellite, elimination ofany need for the thrusters to point through the center of mass of thesatellite, thus reducing the need for dedicated station-keepingthrusters, and the potential to enable completely autonomous orbit andACS control.

One embodiment of the present methods of simultaneous orbit control andmomentum dumping in a spacecraft, the spacecraft including a pluralityof thrusters, comprises the steps of: generating a set of firingcommands for the thrusters from solutions to momentum dumping andinclination control equations; and firing the thrusters according to thefiring commands. The momentum dumping and inclination control equationsare defined as

$\sqrt{{\Delta \; P_{K_{2}}^{2}} + {\Delta \; P_{H_{2}}^{2}}} = {\Delta \; P_{I}}$$\lambda_{Inclination} = {{atan}\; 2( {\frac{\Delta \; P_{H_{2}}}{\Delta \; P_{I}},\frac{\Delta \; P_{K_{2}}}{\Delta \; P_{I}}} )}$${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {C_{{Orbit}\mspace{11mu} {to}\mspace{11mu} {ECI}}\Delta \; \overset{harpoonup}{H}}$${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{normal}\Delta \; t_{i}}} = {\Delta \; P_{I}}};$

where

Δ{right arrow over (H)}=momentum dumping requirement (vector) in orbitframe

Δ{right arrow over (H)}_(ECI)=momentum dumping requirement (vector) inEarth-Centered Inertial frame

ΔP_(K) ₂ =spacecraft mass X minimum delta velocity required to controlmean K₂

ΔP_(H) ₂ =spacecraft mass X minimum delta velocity required to controlmean H₂

{right arrow over (R)}_(i)=lever arm (vector) about the c.g. for thei^(th) thruster in spacecraft body frame

{right arrow over (F)}_(i)=thrust vector for the i^(th) thruster inspacecraft body frame

Δt_(i)=on time for the i^(th) thruster

λ_(Inclination)=location of the maneuver

c_(Orbit to ECI)=transformation matrix from orbit to ECI frame,

-   -   rotation matrix about the Z by λ_(Inclination)

c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe

{right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over (R)}_(i)

${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{11mu} {to}\mspace{11mu} {Orbit}}{\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$

Another embodiment of the present methods of simultaneous orbit controland momentum dumping in a spacecraft, the spacecraft including aplurality of thrusters, comprises the steps of: generating a set offiring commands for the thrusters from solutions to momentum dumping anddrift control equations; and firing the thrusters according to thefiring commands. The momentum dumping and drift control equations aredefined as

${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{tangenial}\Delta \; t_{i}}} - {\Delta \; P_{Drift}}};$

where

Δ{right arrow over (H)}=momentum dumping requirement (vector) in orbitframe

ΔP_(Drift)=spacecraft mass X minimum delta velocity required to controlmean Drift

{right arrow over (R)}_(i)=lever arm (vector) about the c.g. for thei^(th) thruster in spacecraft body frame

{right arrow over (F)}_(i)=thrust vector for the i^(th) thruster inspacecraft body frame

Δt_(i)=on time for the i^(th) thruster

c_(Orbit to ECI)=transformation matrix from orbit to ECI frame

c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe

{right arrow over (r)}_(i)==c_(Body to Orbit) {right arrow over (R)}_(i)

${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{11mu} {to}\mspace{11mu} {Orbit}}{\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$

Another embodiment of the present methods simultaneous orbit control andmomentum dumping in a spacecraft, the spacecraft including a pluralityof thrusters, comprises the steps of: generating a set of firingcommands for the thrusters from solutions to momentum dumping/drift andeccentricity control equations; and firing the thrusters according tothe firing commands. The momentum dumping/drift and eccentricity controlequations are defined as

$P^{tangential} = {\sum\limits_{i}{f_{i}^{tangential}\Delta \; t_{i}}}$$P^{radial} = {\sum\limits_{i}{f_{i}^{radial}\Delta \; t_{i}}}$$\lambda_{Eccentricity} = {\tan^{- 1}( \frac{{2P^{tangential}\Delta \; P_{H_{1}}} + {P^{radial}\Delta \; V_{K_{1}}}}{{2P^{tangential}\Delta \; P_{K_{1}}} - {P^{radial}\Delta \; V_{H_{1}}}} )}$${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {C_{{Orbit}\mspace{11mu} {to}\mspace{11mu} {ECI}}\Delta \; \overset{harpoonup}{H}}$${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{tangential}\Delta \; t_{i}}} = {\Delta \; P_{Drift}}};$

where

Δ{right arrow over (H)}=momentum dumping requirement (vector) in orbitframe

Δ{right arrow over (H)}_(ECI)=momentum dumping requirement (vector) inEarth-Centered Inertial frame

ΔP_(K) ₁ =spacecraft mass X minimum delta velocity required to controlmean K₁

ΔP_(H) ₁ =spacecraft mass X minimum delta velocity required to controlmean H₁

ΔP_(Drift)=spacecraft mass X minimum delta velocity required to controlmean Drift

{right arrow over (R)}_(i)=lever arm (vector) about the c.g. for thei^(th) thruster in spacecraft body frame

{right arrow over (F)}_(i)=thrust vector for the i^(th) thruster inspacecraft body frame

Δt_(i)=on time for the i^(th) thruster

λEccentricity=location of the maneuver

c_(Orbit to ECI)=transformation matrix from orbit to ECI frame,

-   -   rotation matrix about the Z by λ_(Eccentricity)

c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe

{right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over (R)}_(i)

${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{11mu} {to}\mspace{11mu} {Orbit}}{\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$

Another embodiment of the present methods simultaneous orbit control andmomentum dumping in a spacecraft, the spacecraft including a pluralityof thrusters, comprises the steps of: generating a set of firingcommands for the thrusters from solutions to momentum dumping/drift andeccentricity control equations; and firing the thrusters according tothe firing commands. The momentum dumping/drift and eccentricity controlequations are defined as

$\mspace{20mu} {{\sum\limits_{{j = 1},2}P_{j}^{tangential}} = {{{\Delta \; {P_{drift}( {{2P_{1}^{tangential}\cos \; \lambda_{1}} + {P_{1}^{radial}\sin \; \lambda_{1}}} )}} + ( {{2P_{2}^{tangential}{\cos ( {\lambda_{1} - {\Delta\lambda}} )}} + {P_{2}^{radial}{\sin ( {\lambda_{1} - {\Delta\lambda}} )}}} )} = {{{\Delta \; {P_{K_{1}}( {{2P_{1}^{tangential}\sin \; \lambda_{1}} - {P_{1}^{radial}\cos \; \lambda_{1}}} )}} + ( {{2P_{2}^{tangential}{\sin ( {\lambda_{1} - {\Delta\lambda}} )}} - {P_{2}^{radial}{\cos ( {\lambda_{1} - {\Delta\lambda}} )}}} )} = {{{\Delta \; P_{H_{1}}} - {2P_{1}^{radial}P_{2}^{radial}\sin \; {\Delta\lambda}} - {4P_{1}^{tangential}P_{2}^{radial}\cos \; {\Delta\lambda}} - {8P_{1}^{tangential}P_{2}^{tangential}\sin \; {\Delta\lambda}} + {4P_{1}^{radial}P_{2}^{tangential}\cos \; {\Delta\lambda}}} = 0}}}}$  λ₂ = λ₁ − Δλ$\mspace{20mu} {{\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},1}} + {\Delta \; {\overset{harpoonup}{H}}_{{ECI},2}}}}$$\mspace{20mu} {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},1}} = {{C_{{Orbit}\mspace{11mu} {to}\mspace{11mu} {ECI}}( \lambda_{1} )}\Delta \; {\overset{harpoonup}{H}}_{1}}}$$\mspace{20mu} {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},2}} = {{C_{{Orbit}\mspace{11mu} {to}\mspace{11mu} {ECI}}( \lambda_{2} )}\Delta \; {\overset{harpoonup}{H}}_{2}}}$$\mspace{20mu} {P_{j}^{radial} = {\sum\limits_{i}{f_{i,j}^{radial}\Delta \; t_{i,j}}}}$$\mspace{20mu} {P_{j}^{tangential} = {\sum\limits_{i}{f_{i,j}^{tangential}\Delta \; t_{i,j}}}}$$\mspace{20mu} {{{\Delta \; {\overset{harpoonup}{H}}_{j}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i,j} \otimes {\overset{harpoonup}{f}}_{i,j}}\Delta \; t_{i,j}}}};}$  where$\mspace{20mu} {{\overset{harpoonup}{r}}_{i,j} = {C_{{Body}\mspace{11mu} {to}\mspace{11mu} {Orbit}}{\overset{harpoonup}{R}}_{i,j}}}$$\mspace{20mu} {{\overset{harpoonup}{f}}_{i,j} = {{C_{{Body}\mspace{11mu} {to}\mspace{11mu} {Orbit}}{\overset{harpoonup}{F}}_{i,j}} = \begin{bmatrix}f_{i,j}^{tangential} \\f_{i,j}^{radial} \\f_{i,j}^{normal}\end{bmatrix}}}$

One embodiment of the present system for simultaneous orbit control andmomentum dumping of a spacecraft comprises a spacecraft including aplurality of thrusters, and means for generating a set of firingcommands for the thrusters from solutions to momentum dumping andinclination control equations. The momentum dumping and inclinationcontrol equations are defined as

$\sqrt{{\Delta \; P_{K_{2}}^{2}} + {\Delta \; P_{H_{2}}^{2}}} = {\Delta \; P_{I}}$$\lambda_{Inclination} = {{atan}\; 2( {\frac{\Delta \; P_{H_{2}}}{\Delta \; P_{I}},\frac{\Delta \; P_{K_{2}}}{\Delta \; P_{I}}} )}$${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {C_{{Orbit}\mspace{11mu} {to}\mspace{11mu} {ECI}}\Delta \; \overset{harpoonup}{H}}$${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{normal}\Delta \; t_{i}}} = {\Delta \; P_{I}}};$

where

Δ{right arrow over (H)}=momentum dumping requirement (vector) in orbitframe

Δ{right arrow over (H)}_(ECI)=momentum dumping requirement (vector) inEarth-Centered inertial frame

ΔP_(K) ₂ =spacecraft mass X minimum delta velocity required to controlmean K₂

ΔP_(H) ₂ =spacecraft mass X minimum delta velocity required to controlmean H₂

{right arrow over (R)}_(i)=lever arm (vector) about the c.g. for thei^(th) thruster in spacecraft body frame

{right arrow over (F)}_(i)=thrust vector for the i^(th) thruster inspacecraft body frame

Δt_(i)=on time for the i^(th) thruster

λ_(Inclination)=location of the maneuver

c_(Orbit to ECI)=transformation matrix from orbit to ECI frame,

-   -   rotation matrix about the Z by λ_(Inclination)

c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe

{right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over (R)}_(i)

${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{11mu} {to}\mspace{11mu} {Orbit}}{\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$

Another embodiment of the present system for simultaneous orbit controland momentum dumping of a spacecraft comprises a spacecraft configuredto orbit Earth in a geostationary orbit and further configured toautonomously control a position of the spacecraft relative to a fixedpoint on Earth. The spacecraft further comprises a spacecraft body and aplurality of thrusters associated with the spacecraft body. Thespacecraft generates a set of firing commands for the thrusters fromsolutions to momentum dumping and inclination control equations, and thespacecraft fires the thrusters according to the firing commands. Themomentum dumping and inclination control equations are defined as

$\sqrt{{\Delta \; P_{K_{2}}^{2}} + {\Delta \; P_{H_{2}}^{2}}} = {\Delta \; P_{I}}$$\lambda_{Inclination} = {{atan}\; 2( {\frac{\Delta \; P_{H_{2}}}{\Delta \; P_{I}},\frac{\Delta \; P_{K_{2}}}{\Delta \; P_{I}}} )}$${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {C_{{Orbit}\mspace{11mu} {to}\mspace{11mu} {ECI}}\Delta \; \overset{harpoonup}{H}}$${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{normal}\Delta \; t_{i}}} = {\Delta \; P_{I}}};$

where

Δ{right arrow over (H)}=momentum dumping requirement (vector) in orbitframe

Δ{right arrow over (H)}_(ECI)=momentum dumping requirement (vector) inEarth-Centered Inertial frame

ΔP_(K) ₂ =spacecraft mass X minimum delta velocity required to controlmean K₂

ΔP_(H) ₂ =spacecraft mass X minimum delta velocity required to controlmean H₂

{right arrow over (R)}_(i)=lever arm (vector) about the c.g. for thei^(th) thruster in spacecraft body frame

{right arrow over (F)}_(i)=thrust vector for the i^(th) thruster inspacecraft body frame

Δt_(i)=on time for the i^(th) thruster

λ_(Inclination)=location of the maneuver

c_(Orbit to ECI)=transformation matrix from orbit to ECI frame,

-   -   rotation matrix about the Z by λ_(Inclination)

c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe

{right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over (R)}_(i)

${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{11mu} {to}\mspace{11mu} {Orbit}}{\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$

The features, functions, and advantages of the present embodiments canbe achieved independently in various embodiments or may be combined inyet other embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

The embodiments of the present system and methods for simultaneousmomentum dumping and orbit control will now be discussed in detail withan emphasis on highlighting the advantageous features. These embodimentsdepict the novel and non-obvious system and methods shown in theaccompanying drawings, which are for illustrative purposes only. Thesedrawings include the following figures, in which like numerals indicatelike parts:

FIG. 1 is a front perspective view of a geostationary satellite orbitingthe Earth.

DETAILED DESCRIPTION

In describing the present embodiments, the following symbols will beused:

Eccentricity(e) Vector:

Σ=Ω+tan⁻¹(tan(ω)cos(i))

h ₁ =e sin(Σ)

k ₁ =e cos(Σ)

Inclination(i) Vector:

h ₂=sin(i)sin Ω

k ₂=sin(i)cos Ω

Ω=right ascension of ascending node ω=argument of perigeeI=i=inclination of the orbit

${{mean}\mspace{14mu} {drift}\mspace{14mu} {rate}} = \lbrack {{\frac{2\pi}{{Period}_{Nominal}}\sqrt{\frac{a^{3}}{\mu}}} - 1} \rbrack$

a=semi-major axisPeriod_(Nominal)=nominal orbital period of the desired orbitV_(synchronous)=orbital velocity at geosynchronous orbitR_(synchronous)=distance from center of the Earth at geosynchronousorbitΔV_(i)=magnitude of the delta velocity for i^(th) maneuvert_(i)=direction cosine of ΔV_(i) along orbit tangential directionn_(i)=direction cosine of ΔV_(i) along orbit normal directionr_(i)=direction cosine of ΔV_(i) along orbit radial directionλ_(i)=applied delta velocity right ascensionΔV_(lon)=minimum delta velocity required for change of argument oflatitude (*mean longitude)ΔV_(drift)=minimum delta velocity required to control mean semi-majoraxis (*longitudinal drift)ΔV_(K) ₁ =minimum delta velocity required to control mean K₁ΔV_(H) ₁ =minimum delta velocity required to control mean H₁ΔV_(K) ₂ =minimum delta velocity required to control mean K₂ΔV_(H) ₂ =minimum delta velocity required to control mean H₂*For geosynchronous orbit

To control orbit, the size(s) (ΔV) and location(s) (λ) of themaneuver(s) that can correct the orbit must be found. The basic controlequations for drift and eccentricity control are:

${\sum\limits_{{i = 1},2}{\Delta \; V_{i}t_{i}}} = {\Delta \; V_{drift}}$${\sum\limits_{{i = 1},2}{\Delta \; {V_{i}( {{2t_{i}\cos \; \lambda_{i}} + {r_{i}\sin \; \lambda_{i}}} )}}} = {\Delta \; V_{K_{1}}}$${\sum\limits_{{i = 1},2}{\Delta \; {V_{i}( {{2t_{i}\sin \; \lambda_{i}} - {r_{i}\cos \; \lambda_{i}}} )}}} = {\Delta \; V_{H_{1}}}$

And the basic control equations for inclination control are:

ΔV ₃ n ₃ cos λ₃ =ΔV _(K) ₂

ΔV ₃ n ₃ sin λ₃ =ΔV _(H) ₂

Under some circumstances, a set of three burns may be used to controlthe longitudinal drift rate, eccentricity [K₁ H₁], and inclination[K₂H₂] for a satellite in near geo-stationary orbit. From the equationsand symbols above, then:

${\sum\limits_{{i = 1},3}{\Delta \; V_{i}r_{i}}} = {\Delta \; V_{Ion}}$${\sum\limits_{{i = 1},3}{\Delta \; V_{i}t_{i}}} = {\Delta \; V_{drift}}$${\sum\limits_{{i = 1},3}{\Delta \; {V_{i}( {{2t_{i}\cos \; \lambda_{i}} + {r_{i}\sin \; \lambda_{i}}} )}}} = {\Delta \; V_{K_{1}}}$${\sum\limits_{1,3}{\Delta \; {V_{i}( {{2t_{i}\sin \; \lambda_{i}} - {r_{i}\cos \; \lambda_{i}}} )}}} = {\Delta \; V_{H_{1}}}$${\sum\limits_{{i = 1},3}{\Delta \; V_{i}n_{i}\cos \; \lambda_{i}}} = {\Delta \; V_{K_{2}}}$${\sum\limits_{{i = 1},3}{\Delta \; V_{i}n_{i}\sin \; \lambda_{i}}} = {\Delta \; V_{H_{2}}}$

Under some circumstances, however, the longitude equation above may notbe used. For example, after orbit initialization the satellite is at thenominal longitude location. Then only the longitudinal drift may need tobe corrected in order to keep the longitude error to within a desiredrange, such as, for example ±0.05°. Therefore, the remaining fiveequations form the basis for the maneuver calculation. In somesituations these equations cannot be solved analytically. However,careful choices regarding, for example, thruster locations andorientations and satellite configurations can simplify their solutions.

Propellant consumption is sometimes the primary concern for chemicalpropulsion systems. Therefore, station-keeping thrusters may beconfigured specifically either for north/south (inclination control) oreast/west (drift and eccentricity control) maneuvers with minimalunwanted components. Under these conditions the set of equations abovebecomes:

${\sum\limits_{{i = 1},2}{\Delta \; V_{i}t_{i}}} = {\Delta \; V_{drift}}$${\sum\limits_{{i = 1},2}{\Delta \; {V_{i}( {{2t_{i}\cos \; \lambda_{i}} + {r_{i}\sin \; \lambda_{i}}} )}}} = {\Delta \; V_{K_{1}}}$${\sum\limits_{{i = 1},2}{\Delta \; {V_{i}( {{2t_{i}\sin \; \lambda_{i}} - {r_{i}\cos \; \lambda_{i}}} )}}} = {\Delta \; V_{H_{1}}}$Δ V₃n₃cos  λ₃ = Δ V_(K₂) Δ V₃n₃sin  λ₃ = Δ V_(H₂)

with the first three equations above controlling drift and eccentricityand the last two equations controlling inclination.

For maneuver planning, the size(s) (ΔV) and location(s) (λ) of theburn(s) that can correct the orbit according to the selected controlstrategy must be found. For a given ΔV_(drift) and [ΔV_(K1) ΔV_(H1)],one can solve for the two sets of ΔV's and λ's analytically byreformulating the equations for drift and eccentricity control:

$\mspace{20mu} {{\sum\limits_{{i = 1},2}{\Delta \; V_{i}t_{i}}} = {\Delta \; V_{drift}}}$ΔV₁(2t₁cos λ₁ + r₁sin λ₁) + ΔV₂(2t₂cos (λ₁ − Δλ) + r₂sin (λ₁ − Δλ)) = ΔV_(K₁)Δ V₁(2t₁sin  λ₁ − r₁cos  λ₁) + Δ V₂(2t₂sin (λ₁ − Δλ) − r₂cos (λ₁ − Δλ)) = Δ V_(H₁) − 2Δ V₁r₁Δ V₂r₂sin  Δ λ − 4Δ V₁t₁Δ V₂r₂cos  Δ λ − 8Δ V₁t₁Δ V₂t₂sin  Δλ + 4Δ V₁r₁Δ V₂t₂cos  Δλ = 0

where λ₂=λ₁−Δλ

In the equations above there are four possible solutions for ΔV₁, ΔV₂,λ₁ and Δλ:

${\Delta \; V_{1}} = \begin{Bmatrix}{\sqrt{B}\frac{\begin{pmatrix}{16{\frac{\Delta \; V_{drift}t_{1}^{2}t_{2}^{2}}{\sqrt{AB}} + 4}{\frac{\Delta \; V_{drift}t_{1}^{2}r_{2}^{2}}{\sqrt{AB}} +}} \\{{\frac{\Delta \; V_{drift}r_{1}^{2}r_{2}^{2}}{\sqrt{AB}} + 4}{\frac{\Delta \; V_{drift}t_{2}^{2}r_{1}^{2}}{\sqrt{AB}} - t_{2}}}\end{pmatrix}}{( {{4t_{2}^{2}} + r_{1}^{2}} )( {4{\frac{t_{2}^{2}t_{1}}{\sqrt{A}} + \frac{t_{1}r_{2}^{2}}{\sqrt{A}} - t_{2}}} )}} \\{\sqrt{B}\frac{\begin{pmatrix}{16{\frac{\Delta \; V_{drift}t_{1}^{2}t_{2}^{2}}{\sqrt{AB}} + 4}{\frac{\Delta \; V_{drift}t_{1}^{2}r_{2}^{2}}{\sqrt{AB}} +}} \\{{\frac{\Delta \; V_{drift}r_{1}^{2}r_{2}^{2}}{\sqrt{AB}} + 4}{\frac{\Delta \; V_{drift}t_{2}^{2}r_{1}^{2}}{\sqrt{AB}} + t_{2}}}\end{pmatrix}}{( {{4t_{2}^{2}} + r_{1}^{2}} )( {4{\frac{t_{2}^{2}t_{1}}{\sqrt{A}} + \frac{t_{1}r_{2}^{2}}{\sqrt{A}} - t_{2}}} )}} \\{\sqrt{B}\frac{\begin{pmatrix}{16{\frac{\Delta \; V_{drift}t_{1}^{2}t_{2}^{2}}{\sqrt{AB}} + 4}{\frac{\Delta \; V_{drift}t_{1}^{2}r_{2}^{2}}{\sqrt{AB}} +}} \\{{\frac{\Delta \; V_{drift}r_{1}^{2}r_{2}^{2}}{\sqrt{AB}} + 4}{\frac{\Delta \; V_{drift}t_{2}^{2}r_{1}^{2}}{\sqrt{AB}} + t_{2}}}\end{pmatrix}}{( {{4t_{2}^{2}} + r_{1}^{2}} )( {4{\frac{t_{2}^{2}t_{1}}{\sqrt{A}} + \frac{t_{1}r_{2}^{2}}{\sqrt{A}} + t_{2}}} )}} \\{\sqrt{B}\frac{\begin{pmatrix}{16{\frac{\Delta \; V_{drift}t_{1}^{2}t_{2}^{2}}{\sqrt{AB}} + 4}{\frac{\Delta \; V_{drift}t_{1}^{2}r_{2}^{2}}{\sqrt{AB}} +}} \\{{\frac{\Delta \; V_{drift}r_{1}^{2}r_{2}^{2}}{\sqrt{AB}} + 4}{\frac{\Delta \; V_{drift}t_{2}^{2}r_{1}^{2}}{\sqrt{AB}} - t_{2}}}\end{pmatrix}}{( {{4t_{2}^{2}} + r_{1}^{2}} )( {4{\frac{t_{2}^{2}t_{1}}{\sqrt{A}} + \frac{t_{1}r_{2}^{2}}{\sqrt{A}} + t_{2}}} )}}\end{Bmatrix}$ A = 4t₁²r₂² + 4r₁²t₂² + r₁²r₂² + 16t₁²t₂²B = 4t₁²Δ V_(H₁)² + 4t₁²Δ V_(K₁)² + r₁²Δ V_(K₁)² + r₁²Δ V_(H₁)²${\Delta \; V_{2}} = \begin{Bmatrix}\frac{{- \Delta}\; {V_{drift}( {{4\Delta \; V_{drift}t_{1}^{2}} + r_{1}^{2} - {t_{1}\sqrt{B}}} )}}{( {{4t_{2}^{2}} + r_{1}^{2}} )( {4{\frac{t_{2}^{2}t_{1}}{\sqrt{A}} + \frac{t_{1}r_{2}^{2}}{\sqrt{A}} - t_{2}}} )} \\\frac{{- \Delta}\; {V_{drift}( {{4\Delta \; V_{drift}t_{1}^{2}} + r_{1}^{2} + {t_{1}\sqrt{B}}} )}}{( {{4t_{2}^{2}} + r_{1}^{2}} )( {4{\frac{t_{2}^{2}t_{1}}{\sqrt{A}} + \frac{t_{1}r_{2}^{2}}{\sqrt{A}} - t_{2}}} )} \\\frac{\Delta \; {V_{drift}( {{4\Delta \; V_{drift}t_{1}^{2}} + r_{1}^{2} - {t_{1}\sqrt{B}}} )}}{( {{4t_{2}^{2}} + r_{1}^{2}} )( {4{\frac{t_{2}^{2}t_{1}}{\sqrt{A}} + \frac{t_{1}r_{2}^{2}}{\sqrt{A}} + t_{2}}} )} \\\frac{\Delta \; {V_{drift}( {{4\Delta \; V_{drift}t_{1}^{2}} + r_{1}^{2} + {t_{1}\sqrt{B}}} )}}{( {{4t_{2}^{2}} + r_{1}^{2}} )( {4{\frac{t_{2}^{2}t_{1}}{\sqrt{A}} + \frac{t_{1}r_{2}^{2}}{\sqrt{A}} + t_{2}}} )}\end{Bmatrix}$ A = 4t₁²r₂² + 4r₁²t₂² + r₁²r₂² + 16t₁²t₂²B = 4t₁²Δ V_(H₁)² + 4t₁²Δ V_(K₁)² + r₁²Δ V_(K₁)² + r₁²Δ V_(H₁)²$\lambda_{1} = \begin{Bmatrix}{{atan}\; 2( {\frac{{r_{1}\Delta \; V_{K_{1}}} + {2t_{1}\Delta \; V_{H_{1}}}}{\sqrt{B}},\frac{{2t_{1}\Delta \; V_{K_{1}}} - {r_{1}\Delta \; V_{H_{1}}}}{\sqrt{B}}} )} \\{{atan}\; 2( {{- \frac{{r_{1}\Delta \; V_{K_{1}}} + {2t_{1}\Delta \; V_{H_{1}}}}{\sqrt{B}}},{- \frac{{2t_{1}\Delta \; V_{K_{1}}} - {r_{1}\Delta \; V_{H_{1}}}}{\sqrt{B}}}} )} \\{{atan}\; 2( {\frac{{r_{1}\Delta \; V_{K_{1}}} + {2t_{1}\Delta \; V_{H_{1}}}}{\sqrt{B}},\frac{{2t_{1}\Delta \; V_{K_{1}}} - {r_{1}\Delta \; V_{H_{1}}}}{\sqrt{B}}} )} \\{{atan}\; 2( {{- \frac{{r_{1}\Delta \; V_{K_{1}}} + {2t_{1}\Delta \; V_{H_{1}}}}{\sqrt{B}}},{- \frac{{2t_{1}\Delta \; V_{K_{1}}} - {r_{1}\Delta \; V_{H_{1}}}}{\sqrt{B}}}} )}\end{Bmatrix}$ ${\Delta\lambda} = \begin{Bmatrix}{{atan}\; 2( {\frac{2( {{t_{2}r_{1}} - {t_{1}r_{2}}} )}{\sqrt{A}},\frac{{r_{1}r_{2}} + {4t_{1}t_{2}}}{\sqrt{A}}} )} \\{{atan}\; 2( {\frac{2( {{t_{2}r_{1}} - {t_{1}r_{2}}} )}{\sqrt{A}},\frac{{r_{1}r_{2}} + {4t_{1}t_{2}}}{\sqrt{A}}} )} \\{{atan}\; 2( {{- \frac{2( {{t_{2}r_{1}} - {t_{1}r_{2}}} )}{\sqrt{A}}},{- \frac{{r_{1}r_{2}} + {4t_{1}t_{2}}}{\sqrt{A}}}} )} \\{{atan}\; 2( {{- \frac{2( {{t_{2}r_{1}} - {t_{1}r_{2}}} )}{\sqrt{A}}},{- \frac{{r_{1}r_{2}} + {4t_{1}t_{2}}}{\sqrt{A}}}} )}\end{Bmatrix}$

The solution to the above equations that provides the minimum ΔV₁ andΔV₂ is the most advantageous choice, since smaller velocity changesgenerally consume less fuel than larger velocity changes and sincesmaller velocity changes have less potential to create unwanteddisturbances in the satellite's orbit as compared to larger velocitychanges. However, the solution becomes invalid if either ΔV₁ or ΔV₂ isless than zero, which occurs when the magnitude of ΔV_(drift) approachesthe magnitude of [ΔV_(K1) ΔV_(H1)]. In these situations the formulationfor one maneuver can be used, and one set of ΔV and λ control both thedrift and eccentricity:

${\Delta \; V_{1}} = \frac{\Delta \; V_{drift}}{t_{1}}$$\lambda_{1} = {\tan^{- 1}( \frac{{\Delta \; V_{1}2t_{1}\Delta \; V_{H_{1}}} + {\Delta \; V_{1}r_{1}\Delta \; V_{K_{1}}}}{{\Delta \; V_{1}2t_{1}\Delta \; V_{K_{1}}} - {\Delta \; V_{1}r_{1}\Delta \; V_{H_{1}}}} )}$

According to the equations above, the size of the burn is dictated bythe drift collection while the location of the burn is determined by thedirection of the eccentricity correction [ΔV_(K1) ΔV_(H1)] and thein-plane components of the thrust vector [t₁ r₁]. Since ΔV_(drift) doesnot necessarily have the same magnitude as [ΔV_(K1) ΔV_(H1)] the onemaneuver solution may result in either under correction (undershoot) orover correction (overshoot) of the eccentricity perturbation. In suchcases, the difference can be corrected in the next control cycle. For agiven inclination correction [ΔV_(K2) ΔV_(H2)], the solutions for ΔV andλ are very simple:

${\Delta \; V_{I}} = \sqrt{{\Delta \; V_{H_{2}}^{2}} + {\Delta \; V_{K_{2}}^{2}}}$${\Delta \; V_{3}} = \frac{\Delta \; V_{I}}{n_{3}}$$\lambda_{3} = {{atan}\; 2( {\frac{\Delta \; V_{H_{2}}}{\Delta \; V_{I}},\frac{\Delta \; V_{K_{2}}}{\Delta \; V_{I}}} )}$

According to the present embodiments, it is possible to performsimultaneous momentum dumping and orbit control. Benefits achieved bythe present embodiments include a reduction in the number of maneuversneeded to maintain station, increased efficiency in propellant usage,reduction in transients, tighter orbit control, which has the addedbenefit of reducing the antenna pointing budget, a reduction in thenumber of station-keeping thrusters needed aboard the satellite,elimination of any need for the thrusters to point through the center ofmass of the satellite, thus reducing the need for dedicatedstation-keeping thrusters, and the potential to enable completelyautonomous orbit and ACS control.

In the present embodiments, since the equations for orbit control are inthe orbit frame, the momentum dumping requirement is computed in thesame frame:

Δ{right arrow over (H)}=momentum dumping requirement (vector) in orbitframe

Δ{right arrow over (H)}_(ECI)=momentum dumping requirement (vector) inEarth-Centered Inertial frame

{right arrow over (R)}_(i)=lever arm (vector) about the c.g. for thei^(th) thruster in spacecraft body frame

{right arrow over (F)}_(i)=thrust vector for the i^(th) thruster inspacecraft body frame

Δt_(i)=on time for the i^(th) thruster

c_(Orbit to ECI)=transformation matrix from orbit to ECI frame

c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe

{right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over (R)}_(i)

{right arrow over (f)}_(i)=c_(Body to Orbit) {right arrow over (F)}_(i)

${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}\Delta \; \overset{harpoonup}{H}}$

Using the equation for impulse,

{right arrow over (P)}(impulse)={right arrow over(f)}(thrust)Δt(ontime)=M(spacecraft_mass)Δ{right arrow over(V)}(delta_velocity)

the equations for momentum and orbit control can be reformulated intomore convenient forms by multiplying the orbit control equations by thespacecraft mass, which changes very little for small burns:

$\begin{matrix}{{M{\sum\limits_{i}{\Delta \; V_{i}t_{i}}}} = {M\; \Delta \; V_{drift}}} & -> & {{\sum\limits_{i}{f_{i}^{tengential}\Delta \; t_{i}}} = {\Delta \; P_{drift}}} \\{{M{\sum\limits_{i}{\Delta \; {V_{i}( {{2t_{i}\cos \; \lambda_{i}} + {r_{i}\sin \; \lambda_{i}}} )}}}} = {M\; \Delta \; V_{K_{1}}}} & -> & {{\sum\limits_{i}{( {{2f_{i}^{tangential}\cos \; \lambda_{i}} + {f_{i}^{radial}\sin \; \lambda_{i}}} )\Delta \; t_{i}}} = {\Delta \; P_{K_{1}}}} \\{{M{\sum\limits_{i}{\Delta \; {V_{i}( {{2t_{i}\sin \; \lambda_{i}} - {r_{i}\cos \; \lambda_{i}}} )}}}} = {M\; \Delta \; V_{H_{1}}}} & -> & {{\sum\limits_{i}{( {{2f_{i}^{tangential}\sin \; \lambda_{i}} - {f_{i}^{radial}\cos \; \lambda_{i}}} )\Delta \; t_{i}}} = {\Delta \; P_{H_{1}}}} \\{{M{\sum\limits_{i}{\Delta \; V_{i}n_{i}\cos \; \lambda_{i}}}} = {M\; \Delta \; V_{K_{2}}}} & -> & {{\sum\limits_{i}{( {f_{i}^{normal}\cos \; \lambda_{i}} )\Delta \; t_{i}}} = {\Delta \; P_{K_{2}}}} \\{{M{\sum\limits_{i}{\Delta \; V_{i}n_{i}\sin \; \lambda_{i}}}} = {M\; \Delta \; V_{H_{2}}}} & -> & {{\sum\limits_{i}{( {f_{i}^{normal}\sin \; \lambda_{i}} )\Delta \; t_{i}}} = {\Delta \; P_{H_{2}}}} \\{{\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}} & -> & {{\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{P}}_{i}}}}\end{matrix}$

There are eight equations above, five for the orbit control and threefor the momentum dump. Accordingly, the equations require eight unknownsfor their solutions. However, since the orientation of ΔH (the momentumvector in the orbit frame) varies with orbital position of thespacecraft, closed form solutions to the eight equations above can befound by coupling the momentum dumping with orbit control in specificdirections. For example, coupling the momentum dumping with driftcontrol yields the following simple algebraic equations:

${\sum\limits_{i}{f_{i}^{tangential}\Delta \; t_{i}}} = {\Delta \; P_{drift}}$${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{P}}_{i}}}$

And coupling the momentum dumping with inclination control yields thefollowing equations:

$\sqrt{{\Delta \; P_{H_{2}}^{2}} + {\Delta \; P_{K_{2}}^{2}}} = {\Delta \; P_{I}}$${\sum\limits_{i}{f_{i}^{normal}\Delta \; t_{i}}} = {\Delta \; P_{I}}$${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{P}}_{i}}}$

Either set of equations above requires just four unknowns for theirgeneral solutions. For a satellite with fixed thrusters, the unknown canbe chosen as the on time of the thrusters. Therefore, the momentumdumping and the selected orbit control can advantageously beaccomplished by firing thrusters without the need to mount the thrusterson gimbaled platforms. The momentum dump can be performed in conjunctionwith drift control, or in conjunction with inclination control, or acombination of both.

By solving for the location of the maneuver, the complete solution forthe momentum dumping and inclination control can easily be obtained fromthe following equations:

$\sqrt{{\Delta \; P_{K_{2}}^{2}} + {\Delta \; P_{H_{2}}^{2}}} = {\Delta \; P_{I}}$$\lambda_{Inclination} = {a\; \tan \; 2( {\frac{\Delta \; P_{H_{2}}}{\Delta \; P_{I}},\frac{\Delta \; P_{K_{2}}}{\Delta \; P_{I}}} )}$${\Delta \; \overset{harpoonup}{H}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}^{- 1}\Delta \; {\overset{harpoonup}{H}}_{ECI}}$${\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}} = {\Delta \; \overset{harpoonup}{H}}$${\sum\limits_{i}{f_{i}^{normal}\Delta \; t_{i}}} = {\Delta \; P_{I}}$

where

Δ{right arrow over (H)}=momentum dumping requirement (vector) in orbitframe

Δ{right arrow over (H)}_(ECI)=momentum dumping requirement (vector) inEarth-Centered Inertial frame

ΔP_(K) ₂ =spacecraft mass X minimum delta velocity required to controlmean K₂

ΔP_(H) ₂ =spacecraft mass X minimum delta velocity required to controlmean H₂

{right arrow over (R)}_(i)=lever arm (vector) about the c.g. for thei^(th) thruster in spacecraft body frame

{right arrow over (F)}_(i)=thrust vector for the i^(th) thruster inspacecraft body frame

Δt_(i)=on time for the i^(th) thruster

λ_(Inclination)=location of the maneuver

c_(Orbit to ECI)=transformation matrix from orbit to ECI frame,

-   -   rotation matrix about the Z by λ_(Inclination)

c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe

{right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over (R)}_(i)

${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\mspace{14mu} {\overset{harpoonup}{F}}_{i}} = \begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}}$

Since the maneuver to control the drift is independent of location, thecomplete solution for the momentum dumping and drift control can beobtained from the following algebraic equations:

${\sum\limits_{i}{f_{i}^{tangential}\Delta \; t_{i}}} = {\Delta \; P_{Drift}}$${\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}} = {\Delta \; \overset{harpoonup}{H}}$

where

Δ{right arrow over (H)}=momentum dumping requirement (vector) in orbitframe

ΔP_(Drift)=spacecraft mass X minimum delta velocity required to controlmean Drift

{right arrow over (R)}_(i)=lever arm (vector) about the c.g. for thei^(th) thruster in spacecraft body frame

{right arrow over (F)}_(i)=thrust vector for the i^(th) thruster inspacecraft body frame

Δt_(i)=on time for the i^(th) thruster

c_(Orbit to ECI)=transformation matrix from orbit to ECI frame

c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe

{right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over (R)}_(i)

${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\mspace{14mu} {\overset{harpoonup}{F}}_{i}} = \begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}}$

By placing the drift maneuvers in the locations determined by the driftand eccentricity control equations, the momentum dumping can beperformed in conjunction with the eccentricity control. For one maneuverdrift and eccentricity control, λ_(Eccentricity) can be found by simpleiteration (or root searching method) of the following equations:

$P^{radial} = {\sum\limits_{i}{f_{i}^{radial}\Delta \; t_{i}}}$$\lambda_{Eccentricity} = {\tan^{- 1}( \frac{{2\Delta \; P_{Drift}\Delta \; P_{H_{1}}} + {P^{radial}\Delta \; P_{K_{1}}}}{{2\Delta \; P_{Drift}\Delta \; P_{K_{1}}} - {P^{radial}\Delta \; P_{H_{1}}}} )}$${\Delta \; \overset{harpoonup}{H}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}^{{- 1}\;}\mspace{11mu} \Delta \; {\overset{harpoonup}{H}}_{ECI}}$${\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}} = {\Delta \; \overset{harpoonup}{H}}$${\sum\limits_{i}{f_{i}^{tangential}\mspace{11mu} \Delta \; t_{i}}} = {\Delta \; P_{Drift}}$

where

Δ{right arrow over (H)}=momentum dumping requirement (vector) in orbitframe

Δ{right arrow over (H)}_(ECI)=momentum dumping requirement (vector) inEarth-Centered Inertial frame

ΔP_(K) ₁ =spacecraft mass X minimum delta velocity required to controlmean K₁

ΔP_(H) ₁ =spacecraft mass X minimum delta velocity required to controlmean H₁

ΔP_(Drift)=spacecraft mass X minimum delta velocity required to controlmean Drift

{right arrow over (R)}_(i)=lever arm (vector) about the c.g. for thei^(th) thruster in spacecraft body frame

{right arrow over (F)}_(i)=thrust vector for the i^(th) thruster inspacecraft body frame

Δt_(i)=on time for the i^(th) thruster

λ_(Eccentricity)=location of the maneuver

c_(Orbit to ECI)=transformation matrix from orbit to ECI frame,

-   -   rotation matrix about the Z by λ_(Inclination)

c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe

{right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over (R)}_(i)

${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\mspace{14mu} {\overset{harpoonup}{F}}_{i}} = \begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}}$

The solution for the two-maneuvers eccentricity control can be used inconjunction with the equation for momentum and drift control to obtainthe complete solution for momentum dumping and two maneuversdrift/eccentricity control:

${\sum\limits_{{j = 1},2}{{\overset{\_}{f}}_{j}^{radial}\; \Delta \; {\overset{\_}{t}}_{j}}} = {\sum\limits_{{j = 1},2}P_{j}^{radial}}$${\sum\limits_{{j = 1},2}{{\overset{\_}{f}}_{j}^{tangential}\; \Delta \; {\overset{\_}{t}}_{j}}} = {{\sum\limits_{{j = 1},2}P_{j}^{tangential}} = {{{\Delta \; {P_{drift}\begin{pmatrix}{{2P_{1}^{tangential}\cos \; \lambda_{1}} +} \\{P_{1}^{radial}\sin \; \lambda_{1}}\end{pmatrix}}} + \begin{pmatrix}{{2\; P_{2}^{tangential}\cos \; ( {\lambda_{1} - {\Delta \; \lambda}} )} +} \\{P_{2}^{radial}\mspace{11mu} {\sin ( {\lambda_{1} - {\Delta \; \lambda}} )}}\end{pmatrix}} = {{{\Delta \; {P_{K_{1}}\begin{pmatrix}{{2P_{1}^{tangential}\sin \; \lambda_{1}} -} \\{P_{1}^{radial}\cos \; \lambda_{1}}\end{pmatrix}}} + \begin{pmatrix}{{2\; P_{2}^{tangential}\sin \; ( {\lambda_{1} - {\Delta \; \lambda}} )} -} \\{P_{2}^{radial}\mspace{11mu} {\cos ( {\lambda_{1} - {\Delta \; \lambda}} )}}\end{pmatrix}} = {\Delta \; P_{H_{1}}}}}}$ $\begin{matrix}{{{- 2}P_{1}^{radial}P_{2}^{radial}\sin \; \Delta \; \lambda} - {4P_{1}^{tangential}P_{2}^{radial}\mspace{11mu} \cos \; \Delta \; \lambda} -} \\{{8P_{1}^{tangential}P_{2}^{tangential}\sin \; \Delta \; \lambda} + {4\; P_{1}^{radial}P_{2}^{tangential}\mspace{11mu} \cos \; \Delta \; \lambda}}\end{matrix} = 0$ λ₂ = λ₁ − Δλ${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {\sum\limits_{j}{\Delta \; {\overset{harpoonup}{H}}_{{ECI},j}}}$${\Delta \; {\overset{harpoonup}{H}}_{j}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}^{- 1}\; ( \lambda_{j} )\; \Delta \; {\overset{harpoonup}{H}}_{{ECI},j}}$${\sum\limits_{i}{f_{i,j}^{tangential}\; \Delta \; t_{i,j}}} = P_{j}^{tangential}$${\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i,j} \otimes {\overset{harpoonup}{f}}_{i,j}}\Delta \; t_{i,j}}} = {\Delta \; {\overset{harpoonup}{H}}_{j}}$${\overset{harpoonup}{r}}_{i,j} = {C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\mspace{11mu} {\overset{harpoonup}{R}}_{i,j}}$${\overset{harpoonup}{f}}_{i,j} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\mspace{11mu} {\overset{harpoonup}{F}}_{i,j}} = \begin{bmatrix}f_{i,j}^{tangential} \\f_{i,j}^{radial} \\f_{i,j}^{normal}\end{bmatrix}}$ j = 1, 2  index  for  the  maneuversi = index  for  the  i^(th)  thruster

The four sets of equations above (momentum dumping and inclinationcontrol; momentum dumping and drift control; one maneuver drift andeccentricity control; and two maneuvers drift and eccentricity control)can be performed independently, or in various combinations with oneanother. Example combinations include momentum dumping and inclinationcontrol with one maneuver drift and eccentricity control, and momentumdumping and inclination control with two maneuvers drift andeccentricity control. Under certain circumstances, momentum dumping anddrift control may be performed independently in order to maintain thesatellite's longitude. For orbits that do not require control ofinclination, such as, for example, satellites designed for geo-mobilecommunications, either one maneuver drift and eccentricity control ortwo maneuvers drift and eccentricity control may be used to control theorbit drift and eccentricity.

Using the equations described above for simultaneous momentum dumpingand orbit control, substantial benefits can be achieved. For example,the number of maneuvers needed to maintain station can be reduced. Also,station-keeping maneuvers can be performed with a single burn. Each ofthese benefits contributes to increased efficiency in propellant usage,which in turn extends the satellite's lifespan. If desired, singlestation-keeping maneuvers can be broken into segments, or pulses, whichcan be spaced out over multiple burns. In such embodiments, the pulsescan be separated by lesser time intervals as compared to prior artmethods. For example, the elapsed time between pulses may be on theorder of minutes, rather than hours, and may even be less than oneminute.

The present system and methods also enable tighter orbit control, whichhas the added benefit of reducing the antenna pointing budget. Becausestation-keeping maneuvers can be performed with single burns, or withclosely spaced pulsed burns, transients are reduced. The satellite isthus more likely to be on station, even between pulses. Station-keeping,maneuvers can also be performed with a reduced number of station-keepingthrusters aboard the satellite. For example, some maneuvers can beperformed with as little as three or four thrusters.

The present methods also eliminate the need for the thrusters to pointthrough the center of mass of the satellite, which in turn reduces theneed for dedicated station-keeping thrusters. In certain embodiments,however, some thrusters may point through the center of mass. Thepresent methods can also be performed with thrusters that are notpivotable with respect to the satellite, which reduces the complexityand cost of the satellite. In certain embodiments, however, some or allthrusters may be pivotable with respect to the satellite. For example,the thrusters may be mounted on gimbaled platforms.

The present system and methods of simultaneous momentum dumping andorbit control also facilitate completely autonomous orbit and ACScontrol. Satellites are typically controlled from Earth, withstation-keeping commands transmitted from Earth to the satellite. Thepresent methods, however, facilitate elimination of the Earth-boundcontrol center. The satellite itself may monitor its position andtrajectory, generate station-keeping commands on board, and execute thecommands, all without the need for any intervention from Earth.

While the system and methods above have been described as having utilitywith geosynchronous satellites, those of ordinary skill in the art willappreciate that the present system and methods may also be used fororbit control and momentum dumping in satellites in non-geosynchronouscircular and near circular orbits. For example, the present system andmethods may also be used for satellites in non-geosynchronous low Earthorbit (altitude from approximately 100 km to approximately 2,000 km)and/or medium Earth orbit (altitude from approximately 3,000 km toapproximately 25,000+km).

The above description presents the best mode contemplated for carryingout the present system and methods for simultaneous momentum dumping andorbit control, and of the manner and process of making and using them,in such full, clear, concise, and exact terms as to enable any personskilled in the art to which they pertain to make this system and usethese methods. This system and these methods are, however, susceptibleto modifications and alternate constructions from those discussed abovethat are fully equivalent. Consequently, this system and these methodsare not limited to the particular embodiments disclosed. On thecontrary, this system and these methods cover all modifications andalternate constructions coming within the spirit and scope of the systemand methods as generally expressed by the following claims, whichparticularly point out and distinctly claim the subject matter of thesystem and methods.

1. A method of simultaneous orbit control and momentum dumping in aspacecraft the spacecraft including a plurality of thrusters, the methodcomprising the steps of: generating a set of firing commands for thethrusters from solutions to momentum dumping and inclination controlequations; and firing the thrusters according to the firing commands;wherein the momentum dumping and inclination control equations aredefined as$\sqrt{{\Delta \; P_{K_{2}}^{2}} + {\Delta \; P_{H_{2}}^{2}}} = {\Delta \; P_{I}}$$\lambda_{Inclination} = {a\; \tan \; 2( {\frac{\Delta \; P_{H_{2}}}{\Delta \; P_{I}},\frac{\Delta \; P_{K_{2}}}{\Delta \; P_{I}}} )}$${\Delta \; \overset{harpoonup}{H_{ECI}}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}\Delta \; \overset{harpoonup}{H}}$${\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}} = {\Delta \; \overset{harpoonup}{H}}$${{\sum\limits_{i}{f_{i}^{normal}\Delta \; t_{i}}} = {\Delta \; P_{I}}};$where Δ{right arrow over (H)}=momentum dumping requirement (vector) inorbit frame Δ{right arrow over (H)}_(ECI)=momentum dumping requirement(vector) in Earth-Centered Inertial frame ΔP_(K) ₂ =spacecraft mass Xminimum delta velocity required to control mean K₂ ΔP_(H) ₂ =spacecraftmass X minimum delta velocity required to control mean H₂ {right arrowover (R)}_(i)=lever arm (vector) about the c.g. for the i^(th) thrusterin spacecraft body frame {right arrow over (F)}_(i)=thrust vector forthe i^(th) thruster in spacecraft body frame Δt_(i)=on time for thei^(th) thruster λ_(Inclination)=location of the maneuverc_(Orbit to ECI)=transformation matrix from orbit to ECI frame, rotationmatrix about the Z by λ_(Inclination) c_(Body to Orbit)=transformationmatrix from spacecraft body to orbit frame {right arrow over(r)}_(i)=c_(Body to Orbit) {right arrow over (R)}_(i)${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\mspace{11mu} {\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$
 2. The method of claim 1, wherein the thrusters havefixed orientations relative to the spacecraft, such that the thrusterscannot pivot with respect to the spacecraft.
 3. The method of claim 1,wherein the firing commands generate only a single burn in thethrusters.
 4. The method of claim 1, wherein the firing commandsgenerate multiple pulsed burns in the thrusters, and the burns areseparated by intervals of less than thirty minutes.
 5. The method ofclaim 1, wherein the firing commands generate multiple pulsed burns inthe thrusters, and the burns are separated by intervals of less than tenminutes.
 6. The method of claim 1, wherein the firing commands generatemultiple pulsed burns in the thrusters, and the burns are separated byintervals of less than five minutes.
 7. The method of claim 1, whereinwhen each thruster is fired it applies a force to the spacecraft, andnone of said forces points through the center of mass of the spacecraft.8. The method of claim 1, wherein the thrusters are chemical thrustersor ion thrusters.
 9. The method of claim 1, wherein the firing commandsare a first set of firing commands, and further comprising the steps of:generating a second set of firing commands for the thrusters fromsolutions to momentum dumping and drift control equations, and firingthe thrusters according to the second set of firing commands; whereinthe momentum dumping and drift control equations are defined as${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{tangential}\Delta \; t_{i}}} = {\Delta \; P_{Drift}}};$where Δ{right arrow over (H)}=momentum dumping requirement (vector) inorbit frame ΔP_(Drift)=spacecraft mass X minimum delta velocity requiredto control mean Drift {right arrow over (R)}_(i)=lever arm (vector)about the c.g. for the i^(th) thruster in spacecraft body frame {rightarrow over (F)}_(i)=thrust vector for the i^(th) thruster in spacecraftbody frame Δt_(i)=on time for the i^(th) thrusterc_(Orbit to ECI)=transformation matrix from orbit to ECI framec_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe {right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over(R)}_(i)${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\mspace{11mu} {\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$
 10. The method of claim 9, wherein the thrusters arefired according to the first and second sets of firing commandssimultaneously.
 11. The method of claim 10 further comprising the stepof generating a third set of firing commands for the thrusters fromsolutions to momentum dumping/drift and eccentricity control equationswherein the momentum dumping/drift and eccentricity control equationsare defined as$P^{tangential} = {\sum\limits_{i}{f_{i}^{tangential}\Delta \; t_{i}}}$$P^{radial} = {\sum\limits_{i}{f_{i}^{radial}\Delta \; t_{i}}}$$\lambda_{Eccentricity} = {\tan^{- 1}( \frac{{2\; P^{tangential}\Delta \; P_{H_{1}}} + {P^{radial}\Delta \; V_{K_{1}}}}{{2\; P^{tangential}\Delta \; P_{K_{1}}} - {P^{radial}\Delta \; V_{H_{1}}}} )}$${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}^{\;}\mspace{11mu} \Delta \; \overset{harpoonup}{H}}$${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{tangential}\mspace{11mu} \Delta \; t_{i}}} = {\Delta \; P_{Drift}}};$where Δ{right arrow over (H)}=momentum dumping requirement (vector) inorbit frame Δ{right arrow over (H)}_(ECI)=momentum dumping requirement(vector) in Earth-Centered Inertial frame ΔP_(K) ₁ =spacecraft mass Xminimum delta velocity required to control mean K₁ ΔP_(H) ₁ =spacecraftmass X minimum delta velocity required to control mean H₁ΔP_(Drift)=spacecraft mass X minimum delta velocity required to controlmean Drift {right arrow over (R)}_(i)=lever arm (vector) about the c.gfor the i^(th) thruster in spacecraft body frame {right arrow over(F)}_(i)=thrust vector for the i^(th) thruster in spacecraft body frameΔt_(i)=on time for the i^(th) thruster λ_(Eccentricity)=location of themaneuver c_(Orbit to ECI)=transformation matrix from orbit to ECI frame,rotation matrix about the Z by λ_(Eccentricity)c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe {right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over(R)}_(i)${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\mspace{11mu} {\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$
 12. The method of claim 11, wherein the thrusters arefired according to the first, second and third sets of firing commandssimultaneously.
 13. The method of claim 9, further comprising the stepof generating a third set of firing commands for the thrusters fromsolutions to momentum dumping/drift and eccentricity control equationswherein the momentum dumping/drift and eccentricity control equationsare defined as${\sum\limits_{{j = 1},2}P_{j}^{tangential}} = {{{\Delta \; {P_{drift}\begin{pmatrix}{{2P_{1}^{tangential}\cos \; \lambda_{1}} +} \\{P_{1}^{radial}\sin \; \lambda_{1}}\end{pmatrix}}} + \begin{pmatrix}{{2\; P_{2}^{tangential}\cos \; ( {\lambda_{1} - {\Delta \; \lambda}} )} +} \\{P_{2}^{radial}\mspace{11mu} {\sin ( {\lambda_{1} - {\Delta \; \lambda}} )}}\end{pmatrix}} = {{{\Delta \; {P_{K_{1}}\begin{pmatrix}{{2P_{1}^{tangential}\sin \; \lambda_{1}} -} \\{P_{1}^{radial}\cos \; \lambda_{1}}\end{pmatrix}}} + \begin{pmatrix}{{2\; P_{2}^{tangential}\sin \; ( {\lambda_{1} - {\Delta \; \lambda}} )} -} \\{P_{2}^{radial}\mspace{11mu} {\cos ( {\lambda_{1} - {\Delta \; \lambda}} )}}\end{pmatrix}} = {\Delta \; P_{H_{1}}}}}$ $\begin{matrix}{{{- 2}P_{1}^{radial}P_{2}^{radial}\sin \; \Delta \; \lambda} - {4P_{1}^{tangential}P_{2}^{radial}\mspace{11mu} \cos \; \Delta \; \lambda} -} \\{{8P_{1}^{tangential}P_{2}^{tangential}\sin \; \Delta \; \lambda} + {4\; P_{1}^{radial}P_{2}^{tangential}\mspace{11mu} \cos \; \Delta \; \lambda}}\end{matrix} = 0$ λ₂ = λ₁ − Δλ${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},1}} + {\Delta \; {\overset{harpoonup}{H}}_{{ECI},2}}}$${\Delta \; {\overset{harpoonup}{H}}_{{ECI},1}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}\; ( \lambda_{1} )\; \Delta \; {\overset{harpoonup}{H}}_{1}}$${\Delta \; {\overset{harpoonup}{H}}_{{ECI},2}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}\; ( \lambda_{2} )\; \Delta \; {\overset{harpoonup}{H}}_{2}}$$P_{j}^{radial} = {\sum\limits_{i}{f_{i,j}^{radial}\; \Delta \; t_{i,j}}}$$P_{j}^{tangential} = {\sum\limits_{i}{f_{i,j}^{tangential}\; \Delta \; t_{i,j}}}$where${\overset{harpoonup}{r}}_{i,j} = {C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\mspace{11mu} {\overset{harpoonup}{R}}_{i,j}}$${\overset{harpoonup}{f}}_{i,j} = {{C_{{body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\mspace{11mu} {\overset{harpoonup}{F}}_{i,j}} = \begin{bmatrix}f_{i,j}^{tangential} \\f_{i,j}^{radial} \\f_{i,j}^{normal}\end{bmatrix}}$ j = 1, 2  i = index  for  the  i^(th)  thruster.
 14. The method of claim13, wherein the thrusters are fired according to the first, second andthird sets of firing commands simultaneously.
 15. The method of claim 1,wherein the firing commands are a first set of firing commands, andfurther comprising the steps of: generating a second set of firingcommands for the thrusters from solutions to momentum dumping/drift andeccentricity control equations wherein the momentum dumping/drift andeccentricity control equations are defined as$P^{tangential} = {\sum\limits_{i}{f_{i}^{tangential}\Delta \; t_{i}}}$$P^{radial} = {\sum\limits_{i}{f_{i}^{radial}\Delta \; t_{i}}}$$\lambda_{Eccentricity} = {\tan^{- 1}( \frac{{2\; P^{tangential}\Delta \; P_{H_{1}}} + {P^{radial}\Delta \; V_{K_{1}}}}{{2\; P^{tangential}\Delta \; P_{K_{1}}} - {P^{radial}\Delta \; V_{H_{1}}}} )}$${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}^{\;}\mspace{11mu} \Delta \; \overset{harpoonup}{H}}$${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{tangential}\mspace{11mu} \Delta \; t_{i}}} = {\Delta \; P_{Drift}}};$where Δ{right arrow over (H)}=momentum dumping requirement (vector) inorbit frame Δ{right arrow over (H)}_(ECI)=momentum dumping requirement(vector) in Earth-Centered Inertial frame ΔP_(K) ₁ =spacecraft mass Xminimum delta velocity required to control mean K₁ ΔP_(H) ₁ =spacecraftmass X minimum delta velocity required to control mean H₁ΔP_(Drift)=spacecraft mass X minimum delta velocity required to controlmean Drift {right arrow over (R)}_(i)=lever arm (vector) about the c.g.for the i^(th) thruster in spacecraft body frame {right arrow over(F)}_(i)=thrust vector for the i^(th) thruster in spacecraft body frameΔt_(i)=on time for the i^(th) thruster λ_(Eccentricity)=location of themaneuver c_(Orbit to ECI)=transformation matrix from orbit to ECI frame,rotation matrix about the Z by λ_(Eccentricity)c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe {right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over(R)}_(i)${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}{\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$
 16. The method of claim 15, wherein the thrusters arefired according to the first and second sets of firing commandssimultaneously.
 17. The method of claim 1, wherein the firing commandsare a first set of firing commands, and further comprising the steps of:generating a second set of firing commands for the thrusters fromsolutions to momentum dumping/drift and eccentricity control equationswherein the momentum dumping/drift and eccentricity control equationsare defined as$\mspace{20mu} {{\sum\limits_{{j = 1},2}P_{j}^{tangential}} = {{{\Delta \; {P_{drift}( {{2P_{1}^{tangential}\cos \; \lambda_{1}} + {P_{1}^{radial}\sin \; \lambda_{1}}} )}} + ( {{2P_{2}^{tangential}{\cos ( {\lambda_{1} - {\Delta \; \lambda}} )}} + {P_{2}^{radial}{\sin ( {\lambda_{1} - {\Delta\lambda}} )}}} )} = {{{\Delta \; {P_{K_{1}}( {{2P_{1}^{tangential}\sin \; \lambda_{1}} - {P_{1}^{radial}\cos \; \lambda_{1}}} )}} + ( {{2P_{2}^{tangential}{\sin ( {\lambda_{1} - {\Delta \; \lambda}} )}} - {P_{2}^{radial}{\cos ( {\lambda_{1} - {\Delta\lambda}} )}}} )} = {{{\Delta \; P_{H_{1}}} - {2P_{1}^{radial}P_{2}^{radial}\sin \; {\Delta\lambda}} - {4P_{1}^{tangential}P_{2}^{radial}\cos \; \Delta \; \lambda} - {8P_{1}^{tangential}P_{2}^{tangential}\sin \; \Delta \; \lambda} + {4P_{1}^{radial}P_{2}^{tangential}\cos \; \Delta \; \lambda}} = 0}}}}$  λ₂ = λ₁ − Δλ$\mspace{20mu} {{\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},1}} + {\Delta \; {\overset{harpoonup}{H}}_{{ECI},2}}}}$$\mspace{20mu} {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},1}} = {{C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}( \lambda_{1} )}\Delta \; {\overset{harpoonup}{H}}_{1}}}$$\mspace{20mu} {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},2}} = {{C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}( \lambda_{2} )}\Delta \; {\overset{harpoonup}{H}}_{2}}}$$\mspace{20mu} {P_{j}^{radial} = {\sum\limits_{i}{f_{i,j}^{radial}\Delta \; t_{i,j}}}}$$\mspace{20mu} {P_{j}^{tangential} = {\sum\limits_{i}{f_{i,j}^{tangential}\Delta \; t_{i,j}}}}$$\mspace{20mu} {{{\Delta \; {\overset{harpoonup}{H}}_{j}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i,j} \otimes {\overset{harpoonup}{f}}_{i,j}}\Delta \; t_{i,j}}}};}$  where$\mspace{20mu} {{\overset{harpoonup}{r}}_{i,j} = {C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\; {\overset{harpoonup}{R}}_{i,j}}}$$\mspace{20mu} {{\overset{harpoonup}{f}}_{i,j} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}{\overset{harpoonup}{F}}_{i,j}} = \begin{bmatrix}f_{i,j}^{tangential} \\f_{i,j}^{radial} \\f_{i,j}^{normal}\end{bmatrix}}}$   j = 1, 2  i = index  for  the  i^(th)  thruster.
 18. The method ofclaim 17, wherein the thrusters are fired according to the first andsecond sets of firing commands simultaneously.
 19. A method ofsimultaneous orbit control and momentum dumping in a spacecraft, thespacecraft including a plurality of thrusters, the method comprising thesteps of: generating a set of firing commands for the thrusters fromsolutions to momentum dumping and drift control equations; and firingthe thrusters according to the firing commands; wherein the momentumdumping and drift control equations are defined as${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{tangential}\Delta \; t_{i}}} = {\Delta \; P_{Drift}}};$where Δ{right arrow over (H)}=momentum dumping requirement (vector) inorbit frame ΔP_(Drift)=spacecraft mass X minimum delta velocity requiredto control mean Drift {right arrow over (R)}_(i)=lever arm (vector)about the c.g. for the i^(th) thruster in spacecraft body frame {rightarrow over (F)}_(i)=thrust vector for the i^(th) thruster in spacecraftbody frame Δt_(i)=on time for the i^(th) thrusterc_(Orbit to ECI)=transformation matrix from orbit to ECI framec_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe {right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over(R)}_(i)${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}{\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$
 20. The method of claim 19, wherein the thrusters havefixed orientations relative to the spacecraft, such that the thrusterscannot pivot with respect to the spacecraft.
 21. The method of claim 19,wherein the firing commands generate only a single burn in thethrusters.
 22. The method of claim 19 wherein when each thruster isfired it applies a force to the spacecraft, and none of said forcespoints through the center of mass of the spacecraft.
 23. The method ofclaim 19, wherein the thrusters are chemical thrusters or ion thrusters.24. The method of claim 19, wherein the firing commands are a first setof firing commands, and further comprising the step of: generating asecond set of firing commands for the thrusters from solutions tomomentum dumping/drift and eccentricity control equations, wherein themomentum dumping/drift and eccentricity control equations are defined as$P^{tangential} = {\sum\limits_{i}{f_{i}^{tangential}\Delta \; t_{i}}}$$P^{radial} = {\sum\limits_{i}{f_{i}^{radial}\Delta \; t_{i}}}$$\lambda_{Eccentricity} = {\tan^{- 1}( \frac{{2P^{tangential}\Delta \; P_{H_{1}}} + {P^{radial}\Delta \; V_{K_{1}}}}{{2P^{tangential}\Delta \; P_{K_{1}}} - {P^{radial}\Delta \; V_{H_{1}}}} )}$${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}\Delta \; \overset{harpoonup}{H}}$${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{tangential}\Delta \; t_{i}}} = {\Delta \; P_{Drift}}};$where Δ{right arrow over (H)}=momentum dumping requirement (vector) inorbit frame Δ{right arrow over (H)}_(ECI)=momentum dumping requirement(vector) in Earth-Centered Inertial frame ΔP_(K) ₁ =spacecraft mass Xminimum delta velocity required to control mean K₁ ΔP_(H) ₁ =spacecraftmass X minimum delta velocity required to control mean H₁ΔP_(Drift)=spacecraft mass X minimum delta velocity required to controlmean Drift {right arrow over (R)}_(i)=lever arm (vector) about the c.g.for the i^(th) thruster in spacecraft body frame {right arrow over(F)}_(i)=thrust vector for the i^(th) thruster in spacecraft body frameΔt_(i)=on time for the i^(th) thruster λ_(Eccentricity)=location of themaneuver c_(Orbit to ECI)=transformation matrix from orbit to ECI frame,rotation matrix about the Z by λ_(Eccentricity)c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe {right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over(R)}_(i)${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}{\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$
 25. The method of claim 24, wherein the thrusters arefired according to the first and second sets of firing commandssimultaneously.
 26. The method of claim 19, wherein the firing commandsare a first set of firing commands, and further comprising the step of:generating a second set of firing commands for the thrusters fromsolutions to momentum dumping/drift and eccentricity control equationswherein the momentum dumping/drift and eccentricity control equationsare defined as$\mspace{20mu} {{\sum\limits_{{j = 1},2}P_{j}^{tangential}} = {{{\Delta \; {P_{drift}( {{2P_{1}^{tangential}\cos \; \lambda_{1}} + {P_{1}^{radial}\sin \; \lambda_{1}}} )}} + ( {{2P_{2}^{tangential}{\cos ( {\lambda_{1} - {\Delta \; \lambda}} )}} + {P_{2}^{radial}{\sin ( {\lambda_{1} - {\Delta\lambda}} )}}} )} = {{{\Delta \; {P_{K_{1}}( {{2P_{1}^{tangential}\sin \; \lambda_{1}} - {P_{1}^{radial}\cos \; \lambda_{1}}} )}} + ( {{2P_{2}^{tangential}{\sin ( {\lambda_{1} - {\Delta \; \lambda}} )}} - {P_{2}^{radial}{\cos ( {\lambda_{1} - {\Delta\lambda}} )}}} )} = {{{\Delta \; P_{H_{1}}} - {2P_{1}^{radial}P_{2}^{radial}\sin \; {\Delta\lambda}} - {4P_{1}^{tangential}P_{2}^{radial}\cos \; \Delta \; \lambda} - {8P_{1}^{tangential}P_{2}^{tangential}\sin \; \Delta \; \lambda} + {4P_{1}^{radial}P_{2}^{tangential}\cos \; \Delta \; \lambda}} = 0}}}}$  λ₂ = λ₁ − Δλ$\mspace{20mu} {{\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},1}} + {\Delta \; {\overset{harpoonup}{H}}_{{ECI},2}}}}$$\mspace{20mu} {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},1}} = {{C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}( \lambda_{1} )}\Delta \; {\overset{harpoonup}{H}}_{1}}}$$\mspace{20mu} {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},2}} = {{C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}( \lambda_{2} )}\Delta \; {\overset{harpoonup}{H}}_{2}}}$$\mspace{20mu} {P_{j}^{radial} = {\sum\limits_{i}{f_{i,j}^{radial}\Delta \; t_{i,j}}}}$$\mspace{20mu} {P_{j}^{tangential} = {\sum\limits_{i}{f_{i,j}^{tangential}\Delta \; t_{i,j}}}}$$\mspace{20mu} {{{\Delta \; {\overset{harpoonup}{H}}_{j}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i,j} \otimes {\overset{harpoonup}{f}}_{i,j}}\Delta \; t_{i,j}}}};}$  where$\mspace{20mu} {{\overset{harpoonup}{r}}_{i,j} = {C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\; {\overset{harpoonup}{R}}_{i,j}}}$$\mspace{20mu} {{\overset{harpoonup}{f}}_{i,j} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}{\overset{harpoonup}{F}}_{i,j}} = \begin{bmatrix}f_{i,j}^{tangential} \\f_{i,j}^{radial} \\f_{i,j}^{normal}\end{bmatrix}}}$   j = 1, 2  i = index  for  the  i^(th)  thruster.
 27. The method ofclaim 26, wherein the thrusters are fired according to the first andsecond sets of firing commands simultaneously.
 28. A method ofsimultaneous orbit control and momentum dumping in a spacecraft, thespacecraft including a plurality of thrusters the method comprising thesteps of: generating a set of firing commands for the thrusters fromsolutions to momentum dumping/drift and eccentricity control equations;and firing the thrusters according to the firing commands; wherein themomentum dumping/drift and eccentricity control equations are defined as$P^{tangential} = {\sum\limits_{i}{f_{i}^{tangential}\Delta \; t_{i}}}$$P^{radial} = {\sum\limits_{i}{f_{i}^{radial}\Delta \; t_{i}}}$$\lambda_{Eccentricity} = {\tan^{- 1}( \frac{{2P^{tangential}\Delta \; P_{H_{1}}} + {P^{radial}\Delta \; V_{K_{1}}}}{{2P^{tangential}\Delta \; P_{K_{1}}} - {P^{radial}\Delta \; V_{H_{1}}}} )}$${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}\Delta \; \overset{harpoonup}{H}}$${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{tangential}\Delta \; t_{i}}} = {\Delta \; P_{Drift}}};$where Δ{right arrow over (H)}=momentum dumping requirement (vector) inorbit frame Δ{right arrow over (H)}_(ECI)=momentum dumping requirement(vector) in Earth-Centered Inertial frame ΔP_(K) ₁ =spacecraft mass Xminimum delta velocity required to control mean K₁ ΔP_(H) ₁ =spacecraftmass X minimum delta velocity required to control mean H₁ΔP_(Drift)=spacecraft mass X minimum delta velocity required to controlmean Drift {right arrow over (R)}_(i)=lever arm (vector) about the c.g.for the i^(th) thruster in spacecraft body frame {right arrow over(F)}_(i)=thrust vector for the i^(th) thruster in spacecraft body frameΔt_(i)=on time for the i^(th) thruster λ_(Eccentricity)=location of themaneuver c_(Orbit to ECI)=transformation matrix from orbit to ECI frame,rotation matrix about the Z by λ_(Eccentricity)c_(Body to Orbit)=transformation matrix from spacecraft body to orbitframe {right arrow over (r)}_(i)=c_(Body to Orbit) {right arrow over(R)}_(i)${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}{\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$
 29. A method of simultaneous orbit control andmomentum dumping in a spacecraft, the spacecraft including a pluralityof thrusters, the method comprising the steps of: generating a set offiring commands for the thrusters from solutions to momentumdumping/drift and eccentricity control equations; and firing thethrusters according to the firing commands; wherein the momentumdumping/drift and eccentricity control equations are defined as$\mspace{20mu} {{\sum\limits_{{j = 1},2}P_{j}^{tangential}} = {{{\Delta \; {P_{drift}( {{2P_{1}^{tangential}\cos \; \lambda_{1}} + {P_{1}^{radial}\sin \; \lambda_{1}}} )}} + ( {{2P_{2}^{tangential}{\cos ( {\lambda_{1} - {\Delta \; \lambda}} )}} + {P_{2}^{radial}{\sin ( {\lambda_{1} - {\Delta\lambda}} )}}} )} = {{{\Delta \; {P_{K_{1}}( {{2P_{1}^{tangential}\sin \; \lambda_{1}} - {P_{1}^{radial}\cos \; \lambda_{1}}} )}} + ( {{2P_{2}^{tangential}{\sin ( {\lambda_{1} - {\Delta \; \lambda}} )}} - {P_{2}^{radial}{\cos ( {\lambda_{1} - {\Delta\lambda}} )}}} )} = {{{\Delta \; P_{H_{1}}} - {2P_{1}^{radial}P_{2}^{radial}\sin \; {\Delta\lambda}} - {4P_{1}^{tangential}P_{2}^{radial}\cos \; \Delta \; \lambda} - {8P_{1}^{tangential}P_{2}^{tangential}\sin \; \Delta \; \lambda} + {4P_{1}^{radial}P_{2}^{tangential}\cos \; \Delta \; \lambda}} = 0}}}}$  λ₂ = λ₁ − Δλ$\mspace{20mu} {{\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},1}} + {\Delta \; {\overset{harpoonup}{H}}_{{ECI},2}}}}$$\mspace{20mu} {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},1}} = {{C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}( \lambda_{1} )}\Delta \; {\overset{harpoonup}{H}}_{1}}}$$\mspace{20mu} {{\Delta \; {\overset{harpoonup}{H}}_{{ECI},2}} = {{C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}( \lambda_{2} )}\Delta \; {\overset{harpoonup}{H}}_{2}}}$$\mspace{20mu} {P_{j}^{radial} = {\sum\limits_{i}{f_{i,j}^{radial}\Delta \; t_{i,j}}}}$$\mspace{20mu} {P_{j}^{tangential} = {\sum\limits_{i}{f_{i,j}^{tangential}\Delta \; t_{i,j}}}}$$\mspace{20mu} {{{\Delta \; {\overset{harpoonup}{H}}_{j}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i,j} \otimes {\overset{harpoonup}{f}}_{i,j}}\Delta \; t_{i,j}}}};}$  where$\mspace{20mu} {{\overset{harpoonup}{r}}_{i,j} = {C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}\; {\overset{harpoonup}{R}}_{i,j}}}$$\mspace{20mu} {{\overset{harpoonup}{f}}_{i,j} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}{\overset{harpoonup}{F}}_{i,j}} = \begin{bmatrix}f_{i,j}^{tangential} \\f_{i,j}^{radial} \\f_{i,j}^{normal}\end{bmatrix}}}$   j = 1, 2  i = index  for  the  i^(th)  thruster.
 30. A system forsimultaneous orbit control and momentum dumping of a spacecraft,comprising: a spacecraft including a plurality of thrusters; and meansfor generating a set of firing commands for the thrusters from solutionsto momentum dumping and inclination control equations; wherein themomentum dumping and inclination control equations are defined as$\sqrt{{\Delta \; P_{K_{2}}^{2}} + {\Delta \; P_{H_{2}}^{2}}} = {\Delta \; P_{I}}$$\lambda_{Inclination} = {a\; \tan \; 2( {\frac{\Delta \; P_{H_{2}}}{\Delta \; P_{I}},\frac{\Delta \; P_{K_{2}}}{\Delta \; P_{I}}} )}$${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}\Delta \; \overset{harpoonup}{H}}$${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{normal}\Delta \; t_{i}}} = {\Delta \; P_{I}}};$where Δ{right arrow over (H)}=momentum dumping requirement (vector) inorbit frame Δ{right arrow over (H)}_(ECI)=momentum dumping requirement(vector) in Earth-Centered Inertial frame ΔP_(K) ₂ =spacecraft mass Xminimum delta velocity required to control mean K₂ ΔP_(H) ₂ =spacecraftmass X minimum delta velocity required to control mean H₂ {right arrowover (R)}_(i)=lever arm (vector) about the c.g. for the i^(th) thrusterin spacecraft body frame {right arrow over (F)}_(i)=thrust vector forthe i^(th) thruster in spacecraft body frame Δt_(i)=on time for thei^(th) thruster λ_(Inclination)=location of the maneuverc_(Orbit to ECI)=transformation matrix from orbit to ECI frame rotationmatrix about the Z by λ_(Inclination) c_(Body to Orbit)=transformationmatrix from spacecraft body to orbit frame {right arrow over(r)}_(i)=c_(Body to Orbit) {right arrow over (R)}_(i)${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}{\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$
 31. A spacecraft configured to orbit Earth in ageostationary orbit, and further configured to autonomously control aposition of the spacecraft relative to a fixed point on Earth,comprising: a spacecraft body; and a plurality of thrusters associatedwith the spacecraft body; wherein the spacecraft generates a set offiring commands for the thrusters from solutions to momentum dumping andinclination control equations, and the spacecraft fires the thrustersaccording to the firing commands; the momentum dumping and inclinationcontrol equations being defined as$\sqrt{{\Delta \; P_{K_{2}}^{2}} + {\Delta \; P_{H_{2}}^{2}}} = {\Delta \; P_{I}}$$\lambda_{Inclination} = {a\; \tan \; 2( {\frac{\Delta \; P_{H_{2}}}{\Delta \; P_{I}},\frac{\Delta \; P_{K_{2}}}{\Delta \; P_{I}}} )}$${\Delta \; {\overset{harpoonup}{H}}_{ECI}} = {C_{{Orbit}\mspace{14mu} {to}\mspace{14mu} {ECI}}\Delta \; \overset{harpoonup}{H}}$${\Delta \; \overset{harpoonup}{H}} = {\sum\limits_{i}{{{\overset{harpoonup}{r}}_{i} \otimes {\overset{harpoonup}{f}}_{i}}\Delta \; t_{i}}}$${{\sum\limits_{i}{f_{i}^{normal}\Delta \; t_{i}}} = {\Delta \; P_{I}}};$where Δ{right arrow over (H)}momentum dumping requirement (vector) inorbit frame Δ{right arrow over (H)}_(ECI)=momentum dumping requirement(vector) in Earth-Centered Inertial frame ΔP_(K) ₂ =spacecraft mass Xminimum delta velocity required to control mean K₂ ΔP_(H) ₂ =spacecraftmass X minimum delta velocity required to control mean H₂ {right arrowover (R)}_(i)=lever arm (vector) about the c.g. for the i^(th) thrusterin spacecraft body frame {right arrow over (F)}_(i)=thrust vector forthe i^(th) thruster in spacecraft body frame Δt_(i)=on time for thei^(th) thruster λ_(Inclination)=location of the maneuverc_(Orbit to ECI)=transformation matrix from orbit to ECI frame, rotationmatrix about the Z by λ_(Inclination) c_(Body to Orbit)=transformationmatrix from spacecraft body to orbit frame {right arrow over(r)}_(i)=c_(Body to Orbit) {right arrow over (R)}_(i)${\overset{harpoonup}{f}}_{i} = {{C_{{Body}\mspace{14mu} {to}\mspace{14mu} {Orbit}}{\overset{harpoonup}{F}}_{i}} = {\begin{bmatrix}f_{i}^{tangential} \\f_{i}^{radial} \\f_{i}^{normal}\end{bmatrix}.}}$